Modeling of Insurance Data through Two Heavy Tailed Distributions: Computations of Some of Their Actuarial Quantities through Simulation from Their Equilibrium Distributions and the Use of Their Convolutions

Abstract

In this paper, we have fitted two heavy tailed distributions viz the Weibull distribution and the Burr XII distribution to a set of Motor insurance claim data. As it is known, the probability of ruin is obtained as a solution to an integro differential equation, general solution of which leads to what is known as the Pollaczek-Khinchin Formula for the probability of ultimate ruin. In case, the claim severity is distributed as the above two mentioned distributions, and Pollaczek-Khinchin formula cannot be used to evaluate the probability of ruin through inversion of their Laplace transform since the Laplace Transforms themselves don't have closed form expression. However, an approximation to the probability of ultimate ruin in such cases can be obtained by the Pollaczek-Khinchin formula through simulation and one crucial step in this simulation is to simulate from the corresponding Equilibrium distribution of the claim severity distribution. The paper lays down methodologies to simulate from the Equilibrium distribution of Burr XII distribution and Weibull distribution and has used them to obtain an approximation to the probability of ultimate ruin through Pollaczek-Khinchin formula by Monte Carlo simulation. An attempt has also been made to obtain numerical values to the probability function for the number of claims until ruin in case of zero initial surplus under these claim severity distributions and this in turn necessitates the computation of the convolutions of these distributions. The paper makes a preliminary effort to address this issue. All the computations are done under the assumption of the Classical Risk Model.

Analyzing Large Workers’ Compensation Claims Using Generalized Linear Models and Monte Carlo Simulation

“…Using simulation methodology to determine the distribution of the total costs of claims is very beneficial in the field of insurance risk management [36]. Nath and Das [15], Asmussen [37], and Peters et al [38] have applied MC simulation in insurance analyses. According to Hahn [39], insurance companies act as the institutional investors in the financial system of a country and risk dispersion is an important segment of their business.…”

“…According to Boland [12], Achieng [2], and Packová [13], since insurance data holds large infrequent claim amounts, most heavy-tailed distributions can be used to model claim amounts, including gamma, Weibull, exponential, and lognormal distributions. Frees [14] and Nath and Das [15] stated that applying regression models with generalized distributions is useful in modeling skewed and fat-tailed data. Keatinge [16] stated that the exponential distribution gains better results in analyzing loss data.…”

“…Ravi and Butar [17] expressed that heavy-tailed distributions are a much better fit for financial data in comparison to the normal distribution, as financial data are usually highly skewed. Nath and Das [15] applied heavy-tailed distributions (Weibull and Burr) to a set of motor insurance claim data due to the highly skewed nature of the claims. Tang [18] studied the tail behavior of a series of Pareto-type claims.…”